How do I add a 3D border on all edges of a mesh?
One simple way is to duplicate the object (press Shift D, then Esc), enter edit mode, select all and press X to delete Only faces, then add a skin modifier, set the size of the quad border with the "Mean radius" X and Y values of every vertex, then add a subsurf modifier to get the resolution and roundness you need.

Afetr applying all modifiers (always starting from the upper) you can rejoin the two meshes into one single object (Ctrl J).